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 Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 2, Pages 403–411 (Mi tvp294)

Short Communications

Integral Equations and Phase Transitions in Stochastic Games. An Analogy with Statistical Physics

V. P. Maslov

A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Abstract: Maximization of the Kullback–Leibler information is known to result in general Esscher transformations. The Bose–Einstein and Fermi–Dirac statistics in a probability space $(\Omega, \mathcal{F},P)$ give rise to another kind of information, namely,
$$S_B=\int \log(1+\frac{dP}{dQ}) dQ+ \int \log(1+\frac{dQ}{dP}) dP$$
for the Bose statistics and
$$S_F =\int\log(\frac{dP}{dQ}-1) dQ -\int\log(1-\frac{dQ}{dP}) dP, \qquad \frac{dP}{dQ} >1,$$
for the Fermi statistics. This information generates measure transformations corresponding to these statistics. In the presence of a payoff matrix, these transformations vary in accordance with the integral equations given in the paper. We give examples of financial games corresponding to Bose and Fermi statistics.

Keywords: Bose statistics, Fermi statistics, payoff matrix, Esscher transformation, entropy, phase transition, integral equation, Kullback–Leibler information, thermodynamics, statistical physics, dyadic games.

DOI: https://doi.org/10.4213/tvp294

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English version:
Theory of Probability and its Applications, 2004, 48:2, 359–367

Bibliographic databases:

Citation: V. P. Maslov, “Integral Equations and Phase Transitions in Stochastic Games. An Analogy with Statistical Physics”, Teor. Veroyatnost. i Primenen., 48:2 (2003), 403–411; Theory Probab. Appl., 48:2 (2004), 359–367

Citation in format AMSBIB
\Bibitem{Mas03} \by V.~P.~Maslov \paper Integral Equations and Phase Transitions in Stochastic Games. An Analogy with Statistical Physics \jour Teor. Veroyatnost. i Primenen. \yr 2003 \vol 48 \issue 2 \pages 403--411 \mathnet{http://mi.mathnet.ru/tvp294} \crossref{https://doi.org/10.4213/tvp294} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2015462} \zmath{https://zbmath.org/?q=an:1099.91018} \transl \jour Theory Probab. Appl. \yr 2004 \vol 48 \issue 2 \pages 359--367 \crossref{https://doi.org/10.1137/S0040585X97980464} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000222357100013} 

• http://mi.mathnet.ru/eng/tvp294
• https://doi.org/10.4213/tvp294
• http://mi.mathnet.ru/eng/tvp/v48/i2/p403

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This publication is cited in the following articles:
1. V. P. Maslov, “Nonlinear financial averaging, the evolution process, and laws of econophysics”, Theory Probab. Appl., 49:2 (2005), 221–244
2. V. P. Maslov, “Nonlinear Averages in Economics”, Math. Notes, 78:3 (2005), 347–363
3. V. V. V'yugin, V. P. Maslov, “Theorems on Concentration for the Entropy of Free Energy”, Problems Inform. Transmission, 41:2 (2005), 134–149
4. V. V. V'yugin, V. P. Maslov, “Distribution of Investments in the Stock Market, Information Types, and Algorithmic Complexity”, Problems Inform. Transmission, 42:3 (2006), 251–261
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